MATH210: Real Analysis

• Terms Taught: Michaelmas Term Only
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Basic 1st-year mathematics at university level required

Course Description

The notion of a limit underlies a whole range of concepts that are fundamental in mathematics, including sums of infinite series, continuity, differentiation and integration. These ideas belong to the subject of analysis; real analysis considers functions which depend on a real variable.

Following the introductory level treatment given in the first year, our aim now is to further develop a precise understanding of these concepts and to provide proofs of key theorems involving them. We will also show how to apply these theorems in a range of situations.

Educational Aims

On successful completion of this module students should be able to:

· Define the main concepts in real analysis, including convergence of sequences and series, the Riemann integral, continuity of functions, and the derivative

· Construct examples and counter-examples of the concepts listed above

· Apply some common techniques in real analysis to construct elementary proofs using the concepts above

· State some foundational results in real analysis, such as the fundamental theorem of calculus, the intermediate value theorem, and the mean value theorem

Outline Syllabus

Limits of sequences. The Bolzano-Weierstrass theorem. Infinite series and power series. Tests for convergence. Limits and continuity of functions. The intermediate value theorem. Inverse functions. Boundedness and uniform continuity of functions continuous on a closed interval. Sequences and series of functions. Uniform convergence. Differentiation. L'Hopital's Rule. Local maxima and minima. Differentiability of power series. The mean value theorem and its applications; counting zeros of functions. Convex and concave functions Taylor's theorem Riemann integration Applications of integration: inequalities, approximation of finite sums, integrals of convex and concave functions

Assessment Proportions

Assessment will be through

(i) coursework, aimed at testing and consolidating understanding of the basic elements of the course;(ii) an examination in the Summer which assesses more fully the students' understanding and summative knowledge of the topics.

MATH215: Complex Analysis

• Terms Taught: Lent / Summer
• US Credits: 4 semester credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Basic 1st-year mathematics at University level required

Course Description

The purpose of this course is to give an introduction to the theory of functions of a single complex variable, together with some fundamental applications. The treatment will be analytical, and develops ideas from calculus and real analysis. The first part of the course reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The course then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.

Educational Aims

On successful completion of this module, students will be able to:

• understand the ideas of convergence of sequences and series of complex numbers, and of uniform convergence of series of complex functions;
• explain the concepts of continuity and differentiability for complex functions;
• define a line integral and compute line integrals via parametrisation;
• state the fundamental theorem of calculus, several versions of Cauchy's Theorem, Cauchy's Integral Formula for derivatives and Taylor's Theorem, and use these theorem to solve problems;
• explain the notions of zeros and poles for complex functions;
• state Cauchy's estimates, Liouville's Theorem and the fundamental theorem of algebra, and apply these in various situations;
• explain the concept of residues, calculate them, and state and use the Cauchy Residue Theorem;
• apply complex techniques to calculate various real integrals.

Outline Syllabus

• The Argand diagram; polar form for complex numbers.
• Convergence; Cauchy's criterion; uniform convergence and the Weierstrass M test.
• Continuity and differentiability of complex functions; rational functions; differentiability of power series; the exponential function as a power series.
• Line integrals and contours; the fundamental theorem of calculus; Cauchy's theorem for a triangle; Cauchy's formula for a disc.
• Formulae for derivatives; Taylor's theorem; examples.
• Cauchy's theorem for a starlike region; Cauchy's estimates.
• Liouville's theorem; the fundamental theorem of algebra.
• Zeros and poles; the residue theorem and applications.

Assessment Proportions

Assessment will be through:

1. regular coursework, aimed at testing and consolidating understanding of the basic elements of the course;
2. an examination in the Summer, which assesses more fully the students' understanding and summative knowledge of the topics.

MATH220: Linear Algebra II

• Terms Taught: Lent / Summer
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Basic 1st Year mathematics at university level required

Course Description

The explicit aims of this module are to:

• Provide students with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year 2, and also the more specialized year 3 modules.

Educational Aims

On successful completion of this module students will be able to:

• Appreciate the importance of generalization and abstraction in linear algebra
• Follow and to correctly construct mathematical proofs using concepts in linear algebra
• Use linear transformations as a tool for studying and doing calculations with vector spaces, matrices, and linear equations
• Perform calculations of length and angle using the machinery of linear algebra in apparently non-geometric situations, such as the inner product space of matrices or real-valued functions
• Orthogonally diagonalize real symmetric matrices, to find the Jordan normal form of arbitrary matrices, and to recall these techniques in future modules.

Outline Syllabus

• Fields and vector spaces: subspaces, spanning, linear independence, bases, dimension.

• Linear transformations (kernel, image, rank), change of basis
• Inner products, orthogonality, spectral theorem
• Jordan normal form

Assessment Proportions

The lecturer currently plans on having three different forms of coursework:

1. Written coursework, consisting of exercises of a range of difficulty.
2. Weekly online multiple-choice quizzes which assess basic comprehension of the definitions, terminology, and logic
3. Workshop tests, which are short assessed tests in exam-like conditions at the end of the workshops, consisting of a question from a list of questions which the students are given beforehand.

MATH225: Abstract Algebra

• Terms Taught: Michaelmas
• US Credits: 4 semester credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Basic 1st year mathematics at university level required

Course Description

This module has the following subject-specific aims:

• To develop an appreciation of how groups and rings arise as abstract models for studying symmetries and generalizations of number systems;
• To become familiar with a range of examples of groups and rings and to be able to identify important properties of these;
• To study how different groups or rings can be related, through functions between them that preserve the group or ring structure; that is, homomorphisms.
• To demonstrate how the formal logical theories of groups and of rings progress, by developing the basic results in these theories and by proving and applying several fundamental theorems.

Educational Aims

Outline Syllabus

The course will introduce and cover the following topics:

• Permutations, symmetries of two- and three-dimensional shapes and matrices as examples of symmetries;
• Definition of a group, elementary properties;
• Subgroups, cyclic groups and order of elements; cosets; Lagrange's theorem and applications;
• Group homomorphisms and isomorphisms; kernel and image; Cayley's theorem;
• Normal subgroups; quotient groups; the fundamental isomorphism theorem for groups;
• Definition of a ring; examples; rings with identity, commutative rings (including the definition of an integral domain and a field); subrings; ring homomorphisms and isomorphisms;
• Ideals; the ideal structure of the ring of integers;
• Quotient rings; the fundamental isomorphism theorem for rings.

Assessment Proportions

Assessment will be through:

1. regular coursework, aimed at testing and consolidating understanding of the basic elements of the course;
2. an examination in the Summer, which assesses more fully the students' understanding and summative knowledge of the topics.

MATH230: Probability II

• Terms Taught: Lent / Summer
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Basic 1st Year mathematics at University level required

Course Description

This course gives a formal introduction to probability and random variables. Material from the first year on discrete and continuous random variables is revised from a top-down view point. Other commonly used distributions are introduced and key properties proved. Examples from a variety of applications illustrate the theoretical ideas.

The majority of the module aims to extend knowledge of probability and distribution theory so that the student should become competent in manipulating functions of one or more random variables, develop probability models for more realistic problems, and discover how distributions that are important in statistical inference are interlinked. Limits of sequences of random variables will be considered, leading to theoretical results with important practical consequences.

Educational Aims

On successful completion of this module students will be able to:

• Interpret and manipulate the distributions of discrete and continuous univariate and multivariate random variables
• Obtain summary measures such as quantiles, expectation, variance and covariance, of discrete and continuous random variables
• Recognise and relate the distributions of standard random variables
• Identify, with justification, which of the standard probability distributions is likely to be most appropriate for any given application
• Transform random variables and describe how they can be simulated.
• Determine distributional properties of linear combinations of random variables.

Outline Syllabus

• Review of basic probability.
• Random variables and their probability distribution functions. Probability mass functions and probability density function. Quantiles.
• Expectation and variance of random variables; linearity of expectation. Higher-order moments.
• Binomial, Poisson, uniform, geometric exponential. Review from first year and then further properties of the Poisson and exponential.
• The gamma, normal, beta and chi-squared, and their inter-relationships and justification as probability models. The Cauchy and Student-t.
• Joint distribution of vector random variables; that is, systems of two or more random variables, marginal and conditional distributions. Expectations and variances of vector variables. Moment generating functions.
• Properties of linear combinations of random variables.
• Transformations of random variables: motivation, univariate and bivariate methods.
• Limit theory: convergence of variables, laws of large numbers, Central Limit Theorem.
• Multivariate normal distribution.

Assessment Proportions

Assessment will be through:

1. coursework, aimed at testing and consolidating understanding of the basic elements of the course;
2. an examination in the Summer which assesses more fully the students' understanding and summative knowledge of the topics.

MATH235: Statistics II

• Terms Taught: Lent / Summer
• US Credits: 4 US Credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Basic 1st year mathematics at university level required

Course Description

At the end of the module students should be able to:

• appreciate the importance of statistical methodology in making conclusions and decisions.
• recognize the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.
• appreciate the central role of the likelihood function in statistical inference.
• appreciate the role of statistics in making sense of uncertainty.

Educational Aims

On successful completion of this module students will be able to:Critically evaluate whether modelling assumptions are appropriate

• Explain the concept of a sampling distribution as well as properties of estimators
• Construct confidence intervals for estimators, perform hypothesis tests, and appreciate the similarities and differences between the two approaches
• Fit linear regressions using the least squares method to appropriate data
• Write down likelihood functions for simple models and calculate maximum likelihood estimators for parameters
• Use the statistical package 'R' to fit and evaluate models.
• Use statistical procedures to compare different models, and be able to decide which is the most appropriate for a given setting

Outline Syllabus

Statistical methods

• Sampling distributions
• Hypothesis tests

Regression

• Least squares estimaton
• Parameter testing and confidence intervals
• Model comparison
• Model diagnostics
• Model interpretation
• ANOVA as a special case of regression
• Extension of least squares to generalised least squares estimation

• Maximum Likelihood estimation
• Distributions of maximum likelihood estimators; Fisher information
• Confidence intervals of parameters
• Information suppression and sufficiency

Assessment Proportions

Assessment will be through:

1. regular coursework, aimed at testing and consolidating understanding of the basic elements of the course;
2. an examination in the Summer, which assesses more fully the students' understanding and summative knowledge of the topics.

MATH240: Project Skills

• Terms Taught: Michaelmas
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Basic real analysis

Course Description

This module aims to teach and enhance skills, including both subject-related and transferable skills, appropriate to Part II students in Mathematics and Statistics. These skills include the preparation of mathematical documents and presentation materials, scientific writing, oral presentations and group work.

Educational Aims

On successful completion of this module students will be able to

• use LaTeX to prepare a written document;
• use LaTeX to prepare presentational materials;
• create a written report on a mathematical or statistical topic;
• deliver a presentation on a mathematical or statistical topic;
• work in a small group towards researching and presenting a mathematical or statistical topic.

Outline Syllabus

The module consists of 5 components:

• LaTeX. Use of LaTeX to prepare mathematical documents; text and mathematical symbols, displayed formulae, numbering, environments, lists, page and document layout, sections and table of contents, tables and figures, slides for oral presentations.
• Scientific Writing. Style, conventions, good practice, clarity, logical presentation.
• Communication and Presentation Skills. Communication skills, oral communication skills, presenting scientific material verbally, group working.
• Individual Project. Investigation of a mathematical or statistical topic, production of a written report.
• Group Project. Group investigation of a mathematical or statistical topic under the direction of a supervisor, production of a written report, presention of the conclusion.

Assessment Proportions

The assessment for this module is split between the various components:

• LaTeX; coursework (15%).
• Scientific Writing; participation in assigned tasks (5%).
• Communication and Presentation Skills; group oral presentation (15%).
• Individual Project; project (25%).
• Group Project; group project (25%) and oral presentation (15%).

MATH245: Computational Mathematics

• Terms Taught: Lent/Summer
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Basic 1st year mathematics at university level required

Course Description

The aim of this module is to develop the use of computers as a tool for problem solving in mathematics. The module uses the programming language R, which was introduced in the first year, with an emphasis on adopting good programming practices that are transferable to other coding languages and settings. The module includes a substantial project, in which students apply the techniques learned to modelling real-world problems, such as epidemics.

Educational Aims

On successful completion of this module students will be able to:

Write efficient programs in R to solve mathematical problems, using good programming practices.

Understand and implement the process that maps ideas to algorithms to computer code.

Identify and minimise the sources of error in a program or algorithm.

Outline Syllabus

This module will develop students skills in using computers to solve mathematical problems. This includes basic programming techniques, functions and syntax in R, good programming practice, coding simple algorithms, and applications to real-world problems.

Assessment Proportions

Assessment will be through:

Coursework, aimed at testing and consolidating understanding of the basic elements of the course. This will be a mixture of purely formative assessment, online quizzes, and written assessments.

The final summative written assessment will be a substantial project. Students will have time after the end of the taught-section of the course to work on this independently.

The lack of written exam reflects the practical nature of computing which is most appropriately tested via coursework.

MATH4100: Matrices and Calculus

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

MATH4105: Probability and Statistics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

MATH4110: Logic and Discrete Mathematics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

MATH4115: Symmetry and Sequences

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

MATH4120: Mathematical Modelling and Programming

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

MATH4125: Multivariate Calculus

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic 1st - year mathematics at univeristy level required

MATH6310: Metric Spaces and Topology

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic real analysis

MATH6315: Hilbert Spaces

• Terms Taught: Michaelmas term
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

MATH6320: Commutative Algebra

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

MATH6321: Mathematical Cryptography 

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

MATH6325: Representation Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

MATH6326: Graph Theory and Algorithms

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

MATH6327: Knots and Geometry 

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

MATH6330: Statistical Inference

• Terms Taught: Michaelmas 
• US Credits: 5 US Semester credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

MATH6331: Statistical Learning and Prediction

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

MATH6332: Stochastic Processes

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

MATH6333: Bayesian Statistics

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: Basic probability

MATH6335: Medical Statistics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Probability

MATH6337: Environmental Statistics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Probability

MATH6345: Industry-inspired Project

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability / some computational background

MATH6346: Dynamic Modelling

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic computational skills

MATH6347: Mathematics of Generative Modelling

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Linear algebra

MATH6355: Nonlinear Systems and Chaos

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: None

MATH6365: Mathematical Education Placement

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level real analysis

MATH7410: Operators and Spectral Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level real analysis

MATH7415: Measure and Integration

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level (Linear) algebra

MATH7420: Galois Theory

• Terms Taught: Michaelmas 
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

MATH7421: Number Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

MATH7425: Lie Groups and Lie Algebras

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

MATH7426: Combinatorics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability /statistics

MATH7430: Estimation and Inference

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability/ statistics

MATH7431: Advanced Statistical Modelling

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Advanced ststistical Modeling 

MATH7432: Computing and Algorithms for Statistics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability

MATH7434: Probability Theory

• Terms Taught: Lent / Summer 
• US Credits: 5 Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability / statistics

MATH7435: Clinical Trials

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability / statistics

MATH7436: Epidemiology and Disease Modelling

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics

MATH7437: Survival and Longitudinal Statistics

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability.

MATH7439: Stochastic Calculus for Finance

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics.

MATH7445: Hidden-Process Models

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/some computational background.

MATH7446: Machine Learning

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics/ some computational background.